 Hi everyone, this is Alice Gao. In this video, I'm going to discuss the answer to the clicker question slide 11 in lecture 23. In this question, we consider the prisoner's dilemma game, and we want to characterize the Pareto optimal outcome of this game. Turns out three of the four outcomes are Pareto optimal. The only outcome that's not Pareto optimal is the Nash equilibrium and the dominant strategy equilibrium. Which is defect, defect. Let's look at the derivation of this answer. We need to verify that defect is not Pareto optimal, and then we also need to verify the other three outcomes are Pareto optimal. I'm only showing you the reasoning for two of these four cases, but the other two cases are similar to cooperate-cooperate. So first of all, let's verify that defect-defect is not Pareto optimal. Well, how do we verify an outcome is not Pareto optimal? All we have to do is find another outcome that Pareto dominates this outcome. Well, in this game, we can easily see that cooperate-cooperate-cooperate Pareto dominates defect-defect. And the reason is that if you look at both players, both players strictly prefer cooperate-cooperate-cooperate-cooperate-cooperate over defect-defect, because they can get a utility of minus one instead of a utility of minus two. Now, since defect-defect is Pareto dominated by another outcome, so it cannot be Pareto optimal. Okay, and then the reasoning for the other three outcomes are very similar. So let me just show you one of them. Let's look at why cooperate-cooperate is a Pareto optimal outcome. In order to verify that this is a Pareto optimal outcome, we need to verify three things. We need to verify that this outcome is not Pareto dominated by any of the other three outcomes. So we need to verify that defect-cooperate does not Pareto dominate cooperate-cooperate. We need to verify cooperate-defect does not Pareto dominate it. We also need to verify defect-defect does not Pareto dominate it. Let's look at the first case. So the first case, we want to compare defect-cooperate and cooperate-cooperate. Compare these two outcomes and we want to verify that defect-cooperate, so the bottom one does not Pareto dominate the top one. Well, to do that, we just have to find one player who does not weekly prefer the bottom outcome. If we look at Alice, Alice does prefer the bottom outcome, so that doesn't work. But if we look at Bob, for the bottom outcome, Bob gets minus three, for the top one, he gets minus one. So Bob does not prefer the bottom outcome, which means the bottom one does not Pareto dominate the top one. So given this, we know that cooperate-cooperate is not Pareto dominated by defect-cooperate. We can use the same argument for the second one, because this game is symmetric. So cooperate-defect does not Pareto dominate cooperate-cooperate, because Alice does not prefer cooperate-defect. So the reasoning for one and two are mirror images of each other. Now for the third one, third one is the easier case. Why does defect-defect does not Pareto dominate cooperate-cooperate? Well, because both players prefer the other one. So in this case, we're comparing bottom right with top left, and we're trying to verify that the bottom right does not Pareto dominate top left. While both players actually prefer the top left, so therefore bottom right does not Pareto dominate top left. So you can see that verifying an outcome is Pareto optimal is quite a bit of work, because you need to verify that the outcome is not Pareto dominated by any other outcome. Also notice here, we finally have a really good example to show you the nuance of this definition. Notice here, cooperate-cooperate is Pareto optimal, but it does not Pareto dominate all other outcomes. So the definition of Pareto optimality does not require Pareto dominance over all other outcomes. That's a much stronger condition. Instead, we only require that this outcome is not Pareto dominated by any other outcome. That's one important point. And another important point is that you can have two outcomes where neither outcome Pareto dominates the other one. And we have examples here. Let's consider cooperate-cooperate and cooperate-defect. If you look at these two outcomes, Alice prefers the left one, where Bob prefers the right one. In this case, neither outcome Pareto dominates the other outcome. Okay, so this is another nuance of the Pareto dominance relationship. So if we're given two outcomes, it's not necessarily the case that we can always find one outcome to Pareto dominate the other one. Alright, that's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.